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<div>
The research areas of the LMFDB are distinguished by
the variety of connections among the various objects.
</div>


<svg height=600px width=600px viewBox="0 0 1000 1000" xmlns="http://www.w3.org/2000/svg" xlink="http://www.w3.org/1999/xlink" version="1.1">

<g id="AV">
 <text class="AV" x="50" y="130" style="font-family:verdana;font-size:40px">Algebraic</text>
 <text class="AV" x="54" y="175" style="font-family:verdana;font-size:40px">Varieties</text>
</g>

<g id="AF">
 <text class="AF" stroke="none" class="AF" x="620" y="130" style="font-family:verdana;font-size:40px">Automorphic</text>
 <text class="AF" stroke="none" class="AF" x="685" y="175" style="font-family:verdana;font-size:40px">Forms</text>
</g>

<g id="L">
 <text stroke="none" class="L" x="340" y="470" style="font-family:verdana;font-size:40px">L-functions</text>
</g>

<g id="NF">
 <text class="NF" x="69" y="730" style="font-family:verdana;font-size:40px">Number</text>
 <text class="NF" x="88" y="775" style="font-family:verdana;font-size:40px">Fields</text>
</g>

<g id="AR">
 <text stroke="none" class="AR" x="700" y="730" style="font-family:verdana;font-size:35px">Artin</text>
 <text stroke="none" class="AR" x="608" y="770" style="font-family:verdana;font-size:35px">Representations</text>

<svg> <!-- GG and NF to AR -->
<path class="ARkeep" id="GGAR" d="M 380 888 Q 410 793 545 764" stroke="black" stroke-width="2" fill="none"></path>
<path class="ARkeep" id="GGAR" d="M 380 888 l 20,-20 m -20,20 l 20,20" stroke="black" stroke-width="2" fill="none" transform="rotate(-70 380 888)"></path>
<path class="ARkeep" id="NFAR" d="M 270 750 q 160 50 310 5" stroke="black" stroke-width="2" fill="none"></path>
<path class="ARkeep" id="GGAR" d="M 270 750 l 20,-20 m -20,20 l 20,20" stroke="black" stroke-width="2" fill="none" transform="rotate(15 270 750)"></path>
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<text stroke="none" class="AR" style="font-family:ariel;font-size:20px">
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                  &nbsp; &nbsp; component
</textPath>
</text>
<text stroke="none" class="AR" style="font-family:ariel;font-size:20px">
<textPath xlink:href="#arguide2">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; component
</textPath>
</text>
</svg>
</g>

<g id="GG"> <text stroke="none" class="GG" x="325" y="930" style="font-family:verdana;font-size:40px">Galois</text>
 <text stroke="none" class="GG" x="318" y="970" style="font-family:verdana;font-size:40px">Groups</text>
</g>


<svg><!-- arrow GG to NF -->
<defs>
<path id="GGtoNF1" d="M 210 800 L 310 900 l 0 -25 m 0 25 l -25 0" transform="translate(5,-5)"></path>
<path id="GGtoNF2" d="M 210 800 L 310 900 l 0 -25 m 0 25 l -25 0" transform="translate(-15,15)"></path>
</defs>
<g id="NFGG">
  <path class="NFGGkeep" id="GGtoNF" d="M 210 800 L 310 900 l 0 -25 m 0 25 l -25 0" stroke="black" stroke-width="2px" fill="none"></path>

<text stroke="none" class="NFGG" style="font-family:ariel;font-size:24px">
<textPath xlink:href="#GGtoNF1">
&nbsp;&nbsp;Galois
</textPath>
</text>
<text stroke="none" class="NFGG" style="font-family:ariel;font-size:24px">
<textPath xlink:href="#GGtoNF2">
&nbsp;&nbsp;closure
</textPath>
</text>
</g>
<!-- END arrow GG to NF -->
</svg>


<svg><!-- arrow AV to NF -->
<g id="AVNF">
<defs>  
  <path id="AVtoNFoffset" d="M 100 230 L 100 700" stroke="black" stroke-width="0px" fill="none" transform="translate(10,70)"></path>
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<text class="AVNF" style="font-family:ariel;font-size:28px" stroke="none" fill="black">
<textPath xlink:href="#AVtoNFoffset">
field of definition,
</textPath>
</text>
<text class="AVNF" style="font-family:ariel;font-size:28px" stroke="none" fill="black">
<textPath xlink:href="#AVtoNFoffset2">
n-divison fields, etc
</textPath>
</text>
</g>

<!-- END arrow AV to NF -->
</svg>

<svg> <!-- AF lifts -->
<defs>  
  <path id="ARliftoffset" d="M 780,90 A 45,45 0 1,1 825 140" stroke-width="0px" fill="none"></path>
</defs>
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<text id="AFAF" class="AFAF"  style="font-family:ariel;font-size:28px">
<textPath xlink:href="#ARliftoffset">
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;lifts
</textPath>
</text>
</svg>

<svg> <!-- AV lifts -->
<defs>
  <path id="AVtwistoffset" d="M 68,126 A 45,45 0 1,1 138 76" stroke-width="0px" fill="none"></path>
</defs>
 <path d="M 50,145 A 70,70 0 1,1 165,90 l -15 -12  m 15 12 l 12 -15" stroke="steelblue" fill="none" stroke-width="4"></path>

<text id="AVAV" class="AVAV" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#AVtwistoffset">
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;twists
</textPath>
</text>
</svg>

<svg><!-- arrow AR to AF -->
<g id="ARAF">
<defs>
  <path id="ARtoAFoffset" d="M 840 570 L 840 220" stroke-width="0px" fill="none" transform="translate(0,-50)"></path>
</defs>
<path class="ARAF" d="M 850 240 l -20 0 l 20 -20 l 20 20z" stroke="none" stroke-width="1" fill="thistle"></path>
<path class="ARAF" d="M 850 680 l -5 0 l 0 -440 l 10 0 l 0 440z" stroke="none" stroke-width="1" fill="thistle"></path>

<text class="ARAF" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#ARtoAFoffset">
functoriality
</textPath>
</text>
</g>
<!-- END arrow AR to AF -->
</svg>

<svg>  <!-- Number Field to L-function -->
<defs>
  <path id="NFtoLoffsettop" d="M 200 670 L 400 470" stroke="black" stroke-width="0px" fill="none" transform="translate(15,-20)"></path>
  <path id="NFtoLoffsetbottom" d="M 200 670 L 400 470" stroke="black" stroke-width="0px" fill="none" transform="translate(45,15)"></path>
</defs>

<g id="NFL">
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<text class="NFL" style="font-family:ariel;font-size:28px" stroke="none">
<textPath xlink:href="#NFtoLoffsettop">
Dedekind
</textPath>
</text>
<text stroke="none" class="NFL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#NFtoLoffsetbottom">
zeta function
</textPath>
</text>
</g>

</svg> <!-- /Number Field to L-function -->

<svg>  <!-- AV to L-function -->
<g id="AVL">
<defs>
  <path id="AVtoLoffsettop" d="M 200 670 L 400 470" stroke="black" stroke-width="0px" fill="none" transform="rotate(90 250 200) translate(75,-460)"></path>
  <path id="AVtoLoffsetbottom" d="M 200 670 L 400 470" stroke="black" stroke-width="0px" fill="none" transform="rotate(90 250 200) translate(110,-425)"></path>
</defs>

  <path class="AVL" d="M 200 670 L 340 530 L 326 516 L 368 516 L 368 558 L 354 544 L 214 684 Z" transform="rotate(90 250 200) translate(60,-435)" fill="grey"></path>
<text class="AVL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#AVtoLoffsettop">
Hasse-Weil
</textPath>
</text>
<text class="AVL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#AVtoLoffsetbottom">
L-function
</textPath>
</text>
</g>
</svg> <!-- AV to L-function -->

<g id="ARL">
<svg>  <!-- AR to L-function -->
<defs>
  <path id="ARtoLoffsettop" d="M 550 550 L 650 650" stroke-width="0px" fill="none" transform="translate(35,5)"></path>
  <path id="ARtoLoffsetbottom" d="M 550 550 L 650 650" stroke-width="0px" fill="none" transform="translate(-20,25)"></path>
</defs>

  <path class="ARL" d="M 200 670 L 340 530 L 326 516 L 368 516 L 368 558 L 354 544 L 214 684 Z" transform="rotate(-90 250 200) translate(-430,-50)" fill="grey"></path>
<text class="ARL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#ARtoLoffsettop">
Artin
</textPath>
</text>
<text class="ARL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#ARtoLoffsetbottom">
L-function
</textPath>
</text>

</svg> <!-- Artin representation to L-function -->
</g>


<svg>  <!-- L-function to AF  -->
<g id="LAF">
<defs>
  <path id="LtoAFoffsettop" d="M 500 350 l 200 -200" stroke="black" stroke-width="0px" fill="none" transform="translate(30,-35)"></path>
  <path id="LtoAFoffsetbottom" d="M 500 350 l 200 -200" stroke="black" stroke-width="0px" fill="none" transform="translate(67,-7)"></path>
</defs>

  <path class="LAF" d="M 500 350 l 140 -140 l -14 -14 l 42 0 l 0 42 l -14 -14 l -140 140 Z" fill="gold"></path>
<text class="LAF" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#LtoAFoffsettop">
converse
</textPath>
</text>
<text class="LAF" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#LtoAFoffsetbottom">
theorem
</textPath>
</text>
</g>
</svg>

<svg>  <!-- AF to L-function  -->
<g id="AFL">
<defs>
  <path id="AFtoLoffsettop" d="M 470 340 l 200 -200" stroke-width="0px" fill="none" transform="translate(130,55)"></path>
  <path id="AFtoLoffsetbottom" d="M 470 340 l 200 -200" stroke-width="0px" fill="none" transform="translate(165,85)"></path>
</defs>

  <path class="AFL" d="M 470 340 l 140 -140 l -14 -14 l 42 0 l 0 42 l -14 -14 l -140 140 Z" fill="darkorange" transform="rotate(180 610 310)"></path>
<text class="AFL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#AFtoLoffsettop">
automorphic
</textPath>
</text>
<text class="AFL" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#AFtoLoffsetbottom">
representation
</textPath>
</text>
</g>
</svg>

<g id="AVAF">
<svg> <!-- AV to AF -->

<defs>
<path id="modularvarguide" d="M 260 150 q 160 150 320 20" stroke-width="0" fill="none" transform="translate(0,-15)"></path>
</defs>
<path d="M 260 150 q 160 150 320 20" stroke="pink" stroke-width="10" fill="none"></path>
<path d="M 576 176 l -15 -15 l 30 0 l 0 30 Z" fill="pink"></path>
<text class="AVAF" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#modularvarguide">
                  &nbsp; &nbsp; some varieties are modular
</textPath>
</text>
</svg>
</g>

<g id="AFAV">
<svg> <!-- AF to AV -->

<defs>
<path id="svguide" d="M 264 150 q 160 -150 320 10" stroke-width="0" fill="none" transform="translate(0,-15)"></path>
</defs>
<path d="M 260 140 q 160 150 320 20" stroke="lightblue" stroke-width="10" fill="none" transform="rotate(180 425 127)"></path>
<path d="M 576 166 l -15 -15 l 30 0 l 0 30 Z" fill="lightblue" transform="rotate(180 425 127)"></path>
<text class="AFAV" style="font-family:ariel;font-size:28px">
<textPath xlink:href="#svguide">
 &nbsp; &nbsp; &nbsp; &nbsp; 
 &nbsp; &nbsp; Shimura varieties
</textPath>
</text>
</svg>
</g>




<svg>
<path style="display: none;" class="popbox" d="M 500 950 l 0 -100 l 500 0 l 0 100" stroke="none" fill="moccasin" stroke-width="0" display="none"></path>
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<text class="NFhidden" x="510" y="880" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none">A number field is a finite dimensional extension</text>
<text class="NFhidden" x="510" y="905" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none
">of the rational numbers.</text>

<text class="NFLhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The Dedekind zeta function is defined as a</text>
<text class="NFLhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">product over the prime ideals of the number</text>
<text class="NFLhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">field.</text>

<text class="ARhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">An Artin representation is a finite dimensional</text>
<text class="ARhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">representation of the Galois group of a number</text>
<text class="ARhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">field.</text>

<text class="NFGGhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">Associated to a number field is a Galois group:</text>
<text class="NFGGhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">the Galois group of its Galois closure.</text>

<text class="LLhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The Rankin-Selberg convolution of a degree n</text>
<text class="LLhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">L-function and a degree m L-function is an</text>
<text class="LLhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">L-function of degree n*m.</text>

<text class="AVAFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">A variety is 'modular' if its Hasse-Weil</text>
<text class="AVAFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">L-function is the same as the standard  </text>
<text class="AVAFhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">L-function of some modular form.</text>

<text class="AFAVhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">Shimura varieties are algebraic varieties</text>
<text class="AFAVhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size
: 20px; display: none;">uniformized by the same linear algebraic</text>
<text class="AFAVhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">groups that give rise to  automorphic forms.</text>

<text class="AVAVhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">A 'twist' is an algebraic variety defined</text>
<text class="AVAVhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">over the same field isomorphic to the </text>
<text class="AVAVhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">original one over some larger field.</text>

<text class="AFLhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">Given a cuspidal automorphic representation</text>
<text class="AFLhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">&pi; of an adelic reductive group G and a finite </text>
<text class="AFLhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">dimensional representation &rho; of the Weil-</text>
<text class="AFLhidden" x="510" y="955" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">Deligne group of G, one can form the</text>
<text class="AFLhidden" x="510" y="980" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">automorphic L-function L(s,&pi;,&rho;).</text>

<text class="AFAFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">A 'lift' is a map from the automorphic forms  </text>
<text class="AFAFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">on one group to the automorphic forms on  </text>
<text class="AFAFhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">another group.</text>

<text class="LAFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">A 'converse theorem' is an assertion that</text>
<text class="LAFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;"> an L-function is the L-function of an</text>
<text class="LAFhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">automorphic object.</text>

<text class="AVLhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The local factors of the Hasse-Weil</text>
<text class="AVLhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">L-function are determined by counting</text>
<text class="AVLhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">points on the variety modulo prime powers.</text>

<text class="ARLhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The local factors of the Artin L-function are</text>
<text class="ARLhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">constructed from the trace of Frobenius.</text>

<text class="AVhidden" x="510" y="880" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none">Algebraic varieties are geometric objects</text>
<text class="AVhidden" x="510" y="905" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none"> defined by polynomial equations; a more</text>
<text class="AVhidden" x="510" y="930" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none">refined notion is the concept of a scheme.</text>


<text class="AFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">Automorphic forms are L
<tspan baseline-shift="super" style="font-size: 18px">
2
</tspan>
 functions on</text>
<text class="AFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">certain double coset spaces of the adelic</text>
<text class="AFhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">points of a linear algebraic group.</text>

<text class="Lhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">An L-function is a Dirichlet series with a</text>
<text class="Lhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">functional equation and an Euler product.</text>

<text class="GGhidden" x="510" y="880" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none">Galois groups can be realized as permutations</text>
<text class="GGhidden" x="510" y="905" stroke="none" fill="black" style="font-family:verdana;font-size:20px;display:none">acting on the roots of a polynomial.</text>


<text class="ARAFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The Langlands program predicts that Galois</text>
<text class="ARAFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">representations are automorphic.</text>

<text class="AVNFhidden" x="510" y="880" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;"> The &eacute;tale cohomology with torsion coefficients</text>
<text class="AVNFhidden" x="510" y="905" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">carries an action of the absolute Galois group.</text>
<text class="AVNFhidden" x="510" y="930" stroke="none" fill="black" style="font-family: verdana; font-size: 20px; display: none;">The kernel is an n-division field.</text>


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&nbsp;&nbsp;Rankin-Selberg
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